ARMA/ARIMA/SARIMA Models
Recap
In the EDA session, we explore the ACF and PACF for detrended daily stream flow, however, the Augmented Dickey-Fuller Test show that after decomposition, the randomness is still non-stationary.
Log-transform
Code
Figure 1 shows that the data need to be log-transformed. Thus, compared to the analysis in the EDA session, log-transformed data is used here.
ACF and PACF
Code
### can get monthly data
# Get mean value for each month
discharge_df["log_discharge"] <- log(discharge_df["discharge.discharge"])
mean_data <- discharge_df %>%
mutate(month = month(discharge.date), year = year(discharge.date)) %>%
group_by(year, month) %>%
summarize(mean_value = mean(log_discharge))
discharge_month<-ts(mean_data$mean_value,star=decimal_date(as.Date("1970-01-01",format = "%Y-%m-%d")),frequency = 12)
discharge_month2022 <- window(discharge_month,start=1980,end=2022)
decompose_discharge <- decompose(as.ts(discharge_month2022), "additive")
plot(decompose_discharge)
Augmented Dickey-Fuller Test
data: discharge_month2022
Dickey-Fuller = -9.366, Lag order = 7, p-value = 0.01
alternative hypothesis: stationary
The monthly loged data is still not stationary.
Difference
Code
Warning: Removed 3 rows containing missing values (`geom_point()`).
Warning: Removed 3 rows containing missing values (`geom_point()`).
Warning: Removed 3 rows containing missing values (`geom_point()`).
Code
Augmented Dickey-Fuller Test
data: df.ts.discharge1980
Dickey-Fuller = -9.13, Lag order = 7, p-value = 0.01
alternative hypothesis: stationary
Figure 5 shows that the second order differencing may be needed for this data. Figure 4 is not enough, while Figure 6 does not improve much compared to the second order differencing. However, the ADF test show that the time series still not stationary. There might be some seasonal pattern need to be considered. We will go with the second order for now.
For the MA model parameter, q, ACF plot shows that it can be 1 or 2. And the AR model parameter, p, from PACF plot shows that it can be up to 7.
ARIMA model
Code
i=1
d=1
temp= data.frame()
ls=matrix(rep(NA,6*40),nrow=40)
for (p in 1:8)#2 p=0,1,2,3,4,5,6,7
{
for(q in 1:3)#2 q = 0,1,2
{
for(d in 1:2)#2 d=1,2
{
if(p-1+d+q-1<=8)
{
skip_to_next <- FALSE
model<- tryCatch({
Arima(df.ts.discharge1980,order=c(p-1,d,q-1),include.drift=FALSE)
}, error = function(e) { skip_to_next <<- TRUE})
if(skip_to_next) { next }
ls[i,]= c(p-1,d,q-1,model$aic,model$bic,model$aicc)
i=i+1
#print(i)
}
}
}
}
temp= as.data.frame(ls)
names(temp)= c("p","d","q","AIC","BIC","AICc")
#temp
knitr::kable(temp)| p | d | q | AIC | BIC | AICc |
|---|---|---|---|---|---|
| 0 | 1 | 0 | 1357.648 | 1361.846 | 1357.656 |
| 0 | 2 | 0 | 1769.527 | 1773.724 | 1769.535 |
| 0 | 1 | 1 | 1347.007 | 1355.404 | 1347.031 |
| 0 | 2 | 1 | 1364.089 | 1372.482 | 1364.114 |
| 0 | 1 | 2 | 1348.832 | 1361.427 | 1348.881 |
| 0 | 2 | 2 | 1353.817 | 1366.406 | 1353.866 |
| 1 | 1 | 0 | 1346.593 | 1354.989 | 1346.617 |
| 1 | 2 | 0 | 1564.562 | 1572.955 | 1564.586 |
| 1 | 1 | 1 | 1348.447 | 1361.042 | 1348.496 |
| 1 | 2 | 1 | 1353.359 | 1365.948 | 1353.408 |
| 1 | 1 | 2 | 1349.204 | 1365.997 | 1349.286 |
| 1 | 2 | 2 | 1355.176 | 1371.961 | 1355.258 |
| 2 | 1 | 0 | 1348.500 | 1361.095 | 1348.549 |
| 2 | 2 | 0 | 1508.410 | 1521.000 | 1508.460 |
| 2 | 1 | 1 | 1349.326 | 1366.120 | 1349.409 |
| 2 | 2 | 1 | 1355.237 | 1372.022 | 1355.319 |
| 2 | 1 | 2 | 1263.736 | 1284.728 | 1263.859 |
| 2 | 2 | 2 | 1356.069 | 1377.051 | 1356.192 |
| 3 | 1 | 0 | 1349.660 | 1366.453 | 1349.742 |
| 3 | 2 | 0 | 1489.302 | 1506.087 | 1489.384 |
| 3 | 1 | 1 | 1241.035 | 1262.028 | 1241.159 |
| 3 | 2 | 1 | 1356.477 | 1377.459 | 1356.600 |
| 3 | 1 | 2 | 1209.143 | 1234.334 | 1209.316 |
| 3 | 2 | 2 | 1358.036 | 1383.215 | 1358.210 |
| 4 | 1 | 0 | 1346.226 | 1367.218 | 1346.349 |
| 4 | 2 | 0 | 1484.882 | 1505.864 | 1485.005 |
| 4 | 1 | 1 | 1219.709 | 1244.900 | 1219.883 |
| 4 | 2 | 1 | 1353.250 | 1378.429 | 1353.424 |
| 4 | 1 | 2 | 1204.395 | 1233.784 | 1204.626 |
| 4 | 2 | 2 | 1236.064 | 1265.439 | 1236.296 |
| 5 | 1 | 0 | 1329.687 | 1354.878 | 1329.860 |
| 5 | 2 | 0 | 1479.310 | 1504.489 | 1479.484 |
| 5 | 1 | 1 | 1203.909 | 1233.299 | 1204.141 |
| 5 | 2 | 1 | 1337.097 | 1366.472 | 1337.329 |
| 5 | 1 | 2 | 1201.744 | 1235.332 | 1202.042 |
| 6 | 1 | 0 | 1309.947 | 1339.336 | 1310.178 |
| 6 | 2 | 0 | 1477.736 | 1507.111 | 1477.968 |
| 6 | 1 | 1 | 1200.849 | 1234.437 | 1201.147 |
| 7 | 1 | 0 | 1266.364 | 1299.951 | 1266.662 |
| NA | NA | NA | NA | NA | NA |
p d q AIC BIC AICc
38 6 1 1 1200.849 1234.437 1201.147
p d q AIC BIC AICc
38 6 1 1 1200.849 1234.437 1201.147
p d q AIC BIC AICc
33 5 1 1 1203.909 1233.299 1204.141
Code
Coefficients:
ar1 ar2 ar3 ar4 ar5 ar6 ma1 constant
0.5203 0.1032 -0.0665 -0.0945 -0.1351 -0.1040 -1.0000 3e-04
s.e. 0.0448 0.0504 0.0505 0.0505 0.0505 0.0451 0.0067 3e-04
sigma^2 estimated as 0.6407: log likelihood = -591.99, aic = 1201.99
$degrees_of_freedom
[1] 484
$ttable
Estimate SE t.value p.value
ar1 0.5203 0.0448 11.6064 0.0000
ar2 0.1032 0.0504 2.0486 0.0410
ar3 -0.0665 0.0505 -1.3159 0.1888
ar4 -0.0945 0.0505 -1.8719 0.0618
ar5 -0.1351 0.0505 -2.6766 0.0077
ar6 -0.1040 0.0451 -2.3046 0.0216
ma1 -1.0000 0.0067 -148.3871 0.0000
constant 0.0003 0.0003 0.9217 0.3572
$AIC
[1] 2.443064
$AICc
[1] 2.44367
$BIC
[1] 2.519866
Code
Coefficients:
ar1 ar2 ar3 ar4 ar5 ma1 constant
0.5407 0.1139 -0.0596 -0.1062 -0.1910 -1.0000 3e-04
s.e. 0.0442 0.0504 0.0507 0.0505 0.0445 0.0063 4e-04
sigma^2 estimated as 0.648: log likelihood = -594.63, aic = 1205.27
$degrees_of_freedom
[1] 485
$ttable
Estimate SE t.value p.value
ar1 0.5407 0.0442 12.2310 0.0000
ar2 0.1139 0.0504 2.2583 0.0244
ar3 -0.0596 0.0507 -1.1756 0.2403
ar4 -0.1062 0.0505 -2.1033 0.0360
ar5 -0.1910 0.0445 -4.2929 0.0000
ma1 -1.0000 0.0063 -159.2364 0.0000
constant 0.0003 0.0004 0.7963 0.4263
$AIC
[1] 2.44973
$AICc
[1] 2.4502
$BIC
[1] 2.517998
From the results and diagnosis (Figure 7 and Figure 8), ARIMA(5,1,1) and ARIMA(6,1,1) are with lowest AIC, AICc, or BIC. From the diagnosis, both model works well. ARIMA(5,1,1) has less parameters, thus, ARIMA(5,1,1) is chosen as the best model with the equation as follow: \[ x_t = 0.5407*x_{t-1}+0.1139*x_{t-2}-0.0596*x_{t-3}-0.1062*x_{t-4}-0.1910*x_{t-5}-\omega_{t-1}+\omega_t \]
auto.arima()
Using auto.arima() gives results as below. The model is different from the chosen model.The difference comes from that auto.arima() uses different cariteria to chose the best model, i.e.a combination of unit root tests, minimization of the AIC and MLE (source).
Series: df.ts.discharge1980
ARIMA(2,0,2) with non-zero mean
Coefficients:
ar1 ar2 ma1 ma2 mean
1.6474 -0.9023 -1.2396 0.5716 7.8237
s.e. 0.0318 0.0325 0.0754 0.0682 0.0466
sigma^2 = 0.6339: log likelihood = -585.33
AIC=1182.65 AICc=1182.83 BIC=1207.86
Code
Coefficients:
ar1 ar2 ma1 ma2 xmean
1.6474 -0.9023 -1.2396 0.5716 7.8237
s.e. 0.0318 0.0325 0.0754 0.0682 0.0466
sigma^2 estimated as 0.6274: log likelihood = -585.33, aic = 1182.65
$degrees_of_freedom
[1] 488
$ttable
Estimate SE t.value p.value
ar1 1.6474 0.0318 51.8379 0
ar2 -0.9023 0.0325 -27.7802 0
ma1 -1.2396 0.0754 -16.4371 0
ma2 0.5716 0.0682 8.3841 0
xmean 7.8237 0.0466 168.0616 0
$AIC
[1] 2.398894
$AICc
[1] 2.399144
$BIC
[1] 2.450016
Chosen model RMSE
The chosen model is ARIMA(5,1,1).
initial value -0.043422
iter 2 value -0.077676
iter 3 value -0.086688
iter 4 value -0.090021
iter 5 value -0.095887
iter 6 value -0.154643
iter 7 value -0.181523
iter 8 value -0.195392
iter 9 value -0.196261
iter 10 value -0.200343
iter 11 value -0.200970
iter 12 value -0.203454
iter 13 value -0.203627
iter 14 value -0.204076
iter 15 value -0.204154
iter 16 value -0.204176
iter 17 value -0.204177
iter 18 value -0.204178
iter 19 value -0.204178
iter 19 value -0.204178
iter 19 value -0.204178
final value -0.204178
converged
initial value -0.204173
iter 2 value -0.204677
iter 3 value -0.207122
iter 4 value -0.209444
iter 5 value -0.209952
iter 6 value -0.210167
iter 7 value -0.210276
iter 8 value -0.210329
iter 9 value -0.210333
iter 10 value -0.210334
iter 11 value -0.210334
iter 12 value -0.210334
iter 12 value -0.210334
iter 12 value -0.210334
final value -0.210334
converged
[1] 0.8049785
Forecast
Here, using the chosen model ARIMA(5,1,1) to forecast the next 10 months. Figure 10 shows the forecasting plot.
Comparing with benchmark models
In this session, we compared ARIMA model with benchmark models of mean, naive and seasonal naive methods. From the diagnosis in Figure 11, we can see the seasonal naive method is better than the other two. From Summary of ARIMA model and the Seasonal naive model, we can see the chosen ARIMA(5,1,1) has lower RMSE and MAE. However, the seasonal naive method captures better seasonal trend in Figure 12. And the diagnosis plot (Figure 11 (c)) also better than the chosen model in Figure 8.
Code
####################### Benchmark methods ###############################
f1<-meanf(df.ts.discharge1980, h=10) #mean
checkresiduals(f1)#serial correlation ; Lung Box p <0.05
f2<-naive(df.ts.discharge1980, h=10) # naive method
checkresiduals(f2)#serial correlation ; Lung Box p <0.05
f3<-snaive(df.ts.discharge1980, h=10) #seasonal naive method
checkresiduals(f3) #serial correlation ; Lung Box p <0.05
Ljung-Box test
data: Residuals from Mean
Q* = 1216.2, df = 24, p-value < 2.2e-16
Model df: 0. Total lags used: 24
Ljung-Box test
data: Residuals from Naive method
Q* = 157.81, df = 24, p-value < 2.2e-16
Model df: 0. Total lags used: 24
Ljung-Box test
data: Residuals from Seasonal naive method
Q* = 358.05, df = 24, p-value < 2.2e-16
Model df: 0. Total lags used: 24
Code
autoplot(df.ts.discharge1980) +
autolayer(meanf(df.ts.discharge1980, h=10),
series="Mean", PI=FALSE) +
autolayer(naive(df.ts.discharge1980, h=10),
series="Naïve", PI=FALSE) +
autolayer(snaive(df.ts.discharge1980, h=10),
series="Seasonal naïve", PI=FALSE) +
ggtitle("Forecasts for monthly stream flow") +
xlab("Year") + ylab("Log Discharge") +
guides(colour=guide_legend(title="Forecast")) +
autolayer(for_sarima$pred,series="ARIMA(5,1,1)") +
theme(aspect.ratio = 0.4)
#seasonal Naive is goodSummary of ARIMA model
Series: df.ts.discharge1980
ARIMA(5,1,1)
Coefficients:
ar1 ar2 ar3 ar4 ar5 ma1
0.5418 0.1138 -0.0597 -0.1059 -0.1895 -1.0000
s.e. 0.0442 0.0505 0.0507 0.0505 0.0445 0.0092
sigma^2 = 0.6569: log likelihood = -594.95
AIC=1203.91 AICc=1204.14 BIC=1233.3
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 0.004363739 0.8046979 0.6385257 -1.062096 8.377412 0.6903305
ACF1
Training set -0.01913352
Summary of Seasonal naive model
Forecast method: Seasonal naive method
Model Information:
Call: snaive(y = df.ts.discharge1980, h = 10)
Residual sd: 1.1724
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set -0.006577768 1.172446 0.9249566 -1.322599 12.26289 1 0.4562968
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
Feb 2021 9.075799 7.573249 10.578349 6.777847 11.373752
Mar 2021 9.419143 7.916592 10.921693 7.121190 11.717095
Apr 2021 8.459100 6.956550 9.961651 6.161148 10.757053
May 2021 8.599828 7.097278 10.102379 6.301876 10.897781
Jun 2021 7.310991 5.808440 8.813541 5.013038 9.608943
Jul 2021 6.594944 5.092394 8.097495 4.296992 8.892897
Aug 2021 6.224302 4.721751 7.726852 3.926349 8.522254
Sep 2021 6.243943 4.741393 7.746494 3.945991 8.541896
Oct 2021 6.453248 4.950697 7.955798 4.155295 8.751200
Nov 2021 7.269523 5.766973 8.772074 4.971571 9.567476
This is an indication of seasonal effect need to be considered in the model.
SARIMA model
In the last session, ARIMA model was not out performed the benchmark models. This is an indication of that seasonal effect need to be considered in the model. Thus, in this session, SARIMA model is applied to model stream flow time series. From Figure 5, the p1=1, p2=8, q1=1,q2= 3, d1=1,d2=2,Q=1,2, P=1,2.
Code
######################## Check for different combinations ########
#write a funtion
SARIMA.c=function(p1,p2,q1,q2,d1,d2,P1,P2,Q1,Q2,data){
#K=(p2+1)*(q2+1)*(d2)*(P2+1)*(Q2+1)
temp=c()
D=1
s=12
i=1
temp= data.frame()
ls=matrix(rep(NA,9*250),nrow=250)
for (p in p1:p2)
{
for(q in q1:q2)
{
for(d in d1:d2){
for(P in P1:P2)
{
for(Q in Q1:Q2)
{
if(p+d+q+P+Q+D<=12)
{
skip_to_next <- FALSE
model<- tryCatch({
Arima(data,order=c(p-1,d,q-1),seasonal=c(P-1,D,Q-1))
}, error = function(e) { skip_to_next <<- TRUE})
if(skip_to_next) { next }
ls[i,]= c(p-1,d,q-1,P-1,D,Q-1,model$aic,model$bic,model$aicc)
i=i+1
#print(i)
}
}
}
}
}
}
temp= as.data.frame(ls)
names(temp)= c("p","d","q","P","D","Q","AIC","BIC","AICc")
temp
}Code
| p | d | q | P | D | Q | AIC | BIC | AICc |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 0 | 0 | 1 | 0 | 1554.711 | 1558.885 | 1554.720 |
| 0 | 1 | 0 | 0 | 1 | 1 | 1270.686 | 1279.033 | 1270.711 |
| 0 | 1 | 0 | 0 | 1 | 2 | 1272.633 | 1285.154 | 1272.683 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1430.233 | 1438.580 | 1430.258 |
| 0 | 1 | 0 | 1 | 1 | 1 | 1272.634 | 1285.156 | 1272.685 |
| 0 | 1 | 0 | 1 | 1 | 2 | 1273.807 | 1290.502 | 1273.891 |
| 0 | 1 | 0 | 2 | 1 | 0 | 1367.976 | 1380.497 | 1368.026 |
| 0 | 1 | 0 | 2 | 1 | 1 | 1274.519 | 1291.214 | 1274.603 |
| 0 | 1 | 0 | 2 | 1 | 2 | 1276.416 | 1297.285 | 1276.542 |
| 0 | 2 | 0 | 0 | 1 | 0 | 2030.439 | 2034.611 | 2030.448 |
| 0 | 2 | 0 | 0 | 1 | 1 | 1759.319 | 1767.662 | 1759.344 |
| 0 | 2 | 0 | 0 | 1 | 2 | 1761.006 | 1773.522 | 1761.057 |
| 0 | 2 | 0 | 1 | 1 | 0 | 1917.611 | 1925.954 | 1917.636 |
| 0 | 2 | 0 | 1 | 1 | 1 | 1761.031 | 1773.546 | 1761.082 |
| 0 | 2 | 0 | 1 | 1 | 2 | 1761.734 | 1778.421 | 1761.819 |
| 0 | 2 | 0 | 2 | 1 | 0 | 1841.816 | 1854.331 | 1841.867 |
| 0 | 2 | 0 | 2 | 1 | 1 | 1762.431 | 1779.118 | 1762.515 |
| 0 | 2 | 0 | 2 | 1 | 2 | 1763.684 | 1784.543 | 1763.811 |
| 0 | 3 | 0 | 0 | 1 | 0 | 2579.920 | 2584.089 | 2579.928 |
| 0 | 3 | 0 | 0 | 1 | 1 | 2316.845 | 2325.184 | 2316.870 |
| 0 | 3 | 0 | 0 | 1 | 2 | 2318.307 | 2330.816 | 2318.357 |
| 0 | 3 | 0 | 1 | 1 | 0 | 2473.464 | 2481.803 | 2473.489 |
| 0 | 3 | 0 | 1 | 1 | 1 | 2318.364 | 2330.873 | 2318.415 |
| 0 | 3 | 0 | 1 | 1 | 2 | 2318.281 | 2334.960 | 2318.366 |
| 0 | 3 | 0 | 2 | 1 | 0 | 2388.689 | 2401.198 | 2388.739 |
| 0 | 3 | 0 | 2 | 1 | 1 | 2319.315 | 2335.994 | 2319.400 |
| 0 | 3 | 0 | 2 | 1 | 2 | 2319.809 | 2340.657 | 2319.936 |
| 0 | 1 | 1 | 0 | 1 | 0 | 1466.645 | 1474.993 | 1466.670 |
| 0 | 1 | 1 | 0 | 1 | 1 | 1167.104 | 1179.625 | 1167.155 |
| 0 | 1 | 1 | 0 | 1 | 2 | 1168.861 | 1185.556 | 1168.945 |
| 0 | 1 | 1 | 1 | 1 | 0 | 1327.195 | 1339.716 | 1327.246 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1168.861 | 1185.556 | 1168.945 |
| 0 | 1 | 1 | 1 | 1 | 2 | 1170.867 | 1191.736 | 1170.993 |
| 0 | 1 | 1 | 2 | 1 | 0 | 1275.872 | 1292.567 | 1275.956 |
| 0 | 1 | 1 | 2 | 1 | 1 | 1170.860 | 1191.729 | 1170.987 |
| 0 | 1 | 1 | 2 | 1 | 2 | 1172.861 | 1197.903 | 1173.038 |
| 0 | 2 | 1 | 0 | 1 | 0 | 1566.434 | 1574.777 | 1566.459 |
| 0 | 2 | 1 | 0 | 1 | 1 | 1287.852 | 1300.367 | 1287.903 |
| 0 | 2 | 1 | 0 | 1 | 2 | 1289.805 | 1306.492 | 1289.890 |
| 0 | 2 | 1 | 1 | 1 | 0 | 1442.993 | 1455.508 | 1443.044 |
| 0 | 2 | 1 | 1 | 1 | 1 | 1289.807 | 1306.494 | 1289.891 |
| 0 | 2 | 1 | 1 | 1 | 2 | 1290.980 | 1311.838 | 1291.106 |
| 0 | 2 | 1 | 2 | 1 | 0 | 1381.464 | 1398.151 | 1381.548 |
| 0 | 2 | 1 | 2 | 1 | 1 | 1291.683 | 1312.542 | 1291.810 |
| 0 | 2 | 1 | 2 | 1 | 2 | 1293.581 | 1318.611 | 1293.759 |
| 0 | 3 | 1 | 0 | 1 | 0 | 2031.477 | 2039.817 | 2031.503 |
| 0 | 3 | 1 | 0 | 1 | 1 | 1765.597 | 1778.106 | 1765.648 |
| 0 | 3 | 1 | 0 | 1 | 2 | 1767.309 | 1783.987 | 1767.393 |
| 0 | 3 | 1 | 1 | 1 | 0 | 1919.648 | 1932.157 | 1919.699 |
| 0 | 3 | 1 | 1 | 1 | 1 | 1767.332 | 1784.011 | 1767.417 |
| 0 | 3 | 1 | 1 | 1 | 2 | 1768.037 | 1788.885 | 1768.164 |
| 0 | 3 | 1 | 2 | 1 | 0 | 1844.661 | 1861.339 | 1844.745 |
| 0 | 3 | 1 | 2 | 1 | 1 | 1768.698 | 1789.546 | 1768.825 |
| 0 | 1 | 2 | 0 | 1 | 0 | 1456.161 | 1468.682 | 1456.211 |
| 0 | 1 | 2 | 0 | 1 | 1 | 1153.603 | 1170.298 | 1153.687 |
| 0 | 1 | 2 | 0 | 1 | 2 | 1154.982 | 1175.851 | 1155.109 |
| 0 | 1 | 2 | 1 | 1 | 0 | 1320.001 | 1336.696 | 1320.085 |
| 0 | 1 | 2 | 1 | 1 | 1 | 1155.017 | 1175.886 | 1155.143 |
| 0 | 1 | 2 | 1 | 1 | 2 | 1157.603 | 1182.645 | 1157.780 |
| 0 | 1 | 2 | 2 | 1 | 0 | 1256.621 | 1277.490 | 1256.747 |
| 0 | 1 | 2 | 2 | 1 | 1 | 1156.561 | 1181.604 | 1156.739 |
| 0 | 1 | 2 | 2 | 1 | 2 | 1157.240 | 1186.457 | 1157.478 |
| 0 | 2 | 2 | 0 | 1 | 0 | 1480.074 | 1492.589 | 1480.124 |
| 0 | 2 | 2 | 0 | 1 | 1 | 1186.284 | 1202.970 | 1186.368 |
| 0 | 2 | 2 | 0 | 1 | 2 | 1188.093 | 1208.952 | 1188.220 |
| 0 | 2 | 2 | 1 | 1 | 0 | 1341.823 | 1358.510 | 1341.907 |
| 0 | 2 | 2 | 1 | 1 | 1 | 1188.094 | 1208.953 | 1188.221 |
| 0 | 2 | 2 | 1 | 1 | 2 | 1190.109 | 1215.139 | 1190.287 |
| 0 | 2 | 2 | 2 | 1 | 0 | 1291.215 | 1312.074 | 1291.342 |
| 0 | 2 | 2 | 2 | 1 | 1 | 1190.092 | 1215.122 | 1190.270 |
| 0 | 3 | 2 | 0 | 1 | 0 | 1574.826 | 1587.335 | 1574.876 |
| 0 | 3 | 2 | 0 | 1 | 1 | 1294.651 | 1311.330 | 1294.736 |
| 0 | 3 | 2 | 0 | 1 | 2 | 1296.555 | 1317.404 | 1296.683 |
| 0 | 3 | 2 | 1 | 1 | 0 | 1452.387 | 1469.066 | 1452.472 |
| 0 | 3 | 2 | 1 | 1 | 1 | 1296.565 | 1317.413 | 1296.692 |
| 0 | 3 | 2 | 2 | 1 | 0 | 1391.470 | 1412.318 | 1391.597 |
| 0 | 1 | 3 | 0 | 1 | 0 | 1420.254 | 1436.949 | 1420.338 |
| 0 | 1 | 3 | 0 | 1 | 1 | 1127.819 | 1148.688 | 1127.946 |
| 0 | 1 | 3 | 0 | 1 | 2 | 1129.734 | 1154.777 | 1129.912 |
| 0 | 1 | 3 | 1 | 1 | 0 | 1293.812 | 1314.681 | 1293.939 |
| 0 | 1 | 3 | 1 | 1 | 1 | 1129.740 | 1154.783 | 1129.917 |
| 0 | 1 | 3 | 1 | 1 | 2 | 1130.700 | 1159.916 | 1130.937 |
| 0 | 1 | 3 | 2 | 1 | 0 | 1232.385 | 1257.428 | 1232.563 |
| 0 | 1 | 3 | 2 | 1 | 1 | 1131.197 | 1160.413 | 1131.434 |
| 0 | 2 | 3 | 0 | 1 | 0 | 1470.527 | 1487.214 | 1470.611 |
| 0 | 2 | 3 | 0 | 1 | 1 | 1174.511 | 1195.369 | 1174.638 |
| 0 | 2 | 3 | 0 | 1 | 2 | 1176.046 | 1201.077 | 1176.224 |
| 0 | 2 | 3 | 1 | 1 | 0 | 1335.462 | 1356.321 | 1335.589 |
| 0 | 2 | 3 | 1 | 1 | 1 | 1176.075 | 1201.105 | 1176.253 |
| 0 | 2 | 3 | 2 | 1 | 0 | 1273.689 | 1298.720 | 1273.867 |
| 0 | 3 | 3 | 0 | 1 | 0 | 1493.798 | 1510.477 | 1493.883 |
| 0 | 3 | 3 | 0 | 1 | 1 | 1199.306 | 1220.154 | 1199.433 |
| 0 | 3 | 3 | 1 | 1 | 0 | 1356.308 | 1377.156 | 1356.435 |
| 0 | 1 | 4 | 0 | 1 | 0 | 1413.964 | 1434.833 | 1414.090 |
| 0 | 1 | 4 | 0 | 1 | 1 | 1122.773 | 1147.816 | 1122.950 |
| 0 | 1 | 4 | 0 | 1 | 2 | 1124.706 | 1153.923 | 1124.943 |
| 0 | 1 | 4 | 1 | 1 | 0 | 1287.435 | 1312.478 | 1287.612 |
| 0 | 1 | 4 | 1 | 1 | 1 | 1124.710 | 1153.927 | 1124.947 |
| 0 | 1 | 4 | 2 | 1 | 0 | 1227.854 | 1257.071 | 1228.091 |
| 0 | 2 | 4 | 0 | 1 | 0 | 1440.639 | 1461.497 | 1440.766 |
| 0 | 2 | 4 | 0 | 1 | 1 | 1152.585 | 1177.615 | 1152.763 |
| 0 | 2 | 4 | 1 | 1 | 0 | 1315.154 | 1340.184 | 1315.332 |
| 0 | 3 | 4 | 0 | 1 | 0 | 1485.402 | 1506.250 | 1485.529 |
| 0 | 1 | 5 | 0 | 1 | 0 | 1414.965 | 1440.007 | 1415.142 |
| 0 | 1 | 5 | 0 | 1 | 1 | 1121.012 | 1150.228 | 1121.249 |
| 0 | 1 | 5 | 1 | 1 | 0 | 1284.238 | 1313.454 | 1284.475 |
| 0 | 2 | 5 | 0 | 1 | 0 | 1433.850 | 1458.881 | 1434.028 |
| 0 | 1 | 6 | 0 | 1 | 0 | 1414.000 | 1443.217 | 1414.237 |
| 1 | 1 | 0 | 0 | 1 | 0 | 1494.539 | 1502.887 | 1494.565 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1198.259 | 1210.780 | 1198.309 |
| 1 | 1 | 0 | 0 | 1 | 2 | 1200.169 | 1216.864 | 1200.253 |
| 1 | 1 | 0 | 1 | 1 | 0 | 1357.988 | 1370.509 | 1358.038 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1200.169 | 1216.864 | 1200.253 |
| 1 | 1 | 0 | 1 | 1 | 2 | 1202.123 | 1222.992 | 1202.249 |
| 1 | 1 | 0 | 2 | 1 | 0 | 1309.527 | 1326.222 | 1309.611 |
| 1 | 1 | 0 | 2 | 1 | 1 | 1202.168 | 1223.037 | 1202.294 |
| 1 | 1 | 0 | 2 | 1 | 2 | 1204.121 | 1229.164 | 1204.299 |
| 1 | 2 | 0 | 0 | 1 | 0 | 1802.468 | 1810.812 | 1802.493 |
| 1 | 2 | 0 | 0 | 1 | 1 | 1513.828 | 1526.343 | 1513.878 |
| 1 | 2 | 0 | 0 | 1 | 2 | 1515.824 | 1532.511 | 1515.909 |
| 1 | 2 | 0 | 1 | 1 | 0 | 1674.889 | 1687.404 | 1674.939 |
| 1 | 2 | 0 | 1 | 1 | 1 | 1515.824 | 1532.511 | 1515.909 |
| 1 | 2 | 0 | 1 | 1 | 2 | 1517.828 | 1538.686 | 1517.954 |
| 1 | 2 | 0 | 2 | 1 | 0 | 1624.087 | 1640.774 | 1624.171 |
| 1 | 2 | 0 | 2 | 1 | 1 | 1517.731 | 1538.589 | 1517.857 |
| 1 | 2 | 0 | 2 | 1 | 2 | 1519.821 | 1544.851 | 1519.999 |
| 1 | 3 | 0 | 0 | 1 | 0 | 2224.197 | 2232.536 | 2224.223 |
| 1 | 3 | 0 | 0 | 1 | 1 | 1942.277 | 1954.786 | 1942.327 |
| 1 | 3 | 0 | 0 | 1 | 2 | 1944.162 | 1960.841 | 1944.247 |
| 1 | 3 | 0 | 1 | 1 | 0 | 2102.401 | 2114.910 | 2102.452 |
| 1 | 3 | 0 | 1 | 1 | 1 | 1944.166 | 1960.845 | 1944.251 |
| 1 | 3 | 0 | 1 | 1 | 2 | 1946.173 | 1967.021 | 1946.300 |
| 1 | 3 | 0 | 2 | 1 | 0 | 2045.283 | 2061.961 | 2045.367 |
| 1 | 3 | 0 | 2 | 1 | 1 | 1945.994 | 1966.842 | 1946.122 |
| 1 | 1 | 1 | 0 | 1 | 0 | 1414.035 | 1426.557 | 1414.086 |
| 1 | 1 | 1 | 0 | 1 | 1 | 1124.142 | 1140.837 | 1124.227 |
| 1 | 1 | 1 | 0 | 1 | 2 | 1126.113 | 1146.982 | 1126.239 |
| 1 | 1 | 1 | 1 | 1 | 0 | 1287.709 | 1304.404 | 1287.793 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1126.114 | 1146.983 | 1126.241 |
| 1 | 1 | 1 | 1 | 1 | 2 | 1126.707 | 1151.750 | 1126.885 |
| 1 | 1 | 1 | 2 | 1 | 0 | 1224.966 | 1245.835 | 1225.093 |
| 1 | 1 | 1 | 2 | 1 | 1 | 1127.752 | 1152.795 | 1127.930 |
| 1 | 1 | 1 | 2 | 1 | 2 | 1129.615 | 1158.832 | 1129.852 |
| 1 | 2 | 1 | 0 | 1 | 0 | 1506.986 | 1519.501 | 1507.036 |
| 1 | 2 | 1 | 0 | 1 | 1 | 1216.317 | 1233.004 | 1216.402 |
| 1 | 2 | 1 | 0 | 1 | 2 | 1218.253 | 1239.111 | 1218.380 |
| 1 | 2 | 1 | 1 | 1 | 0 | 1371.560 | 1388.247 | 1371.645 |
| 1 | 2 | 1 | 1 | 1 | 1 | 1218.253 | 1239.111 | 1218.380 |
| 1 | 2 | 1 | 1 | 1 | 2 | 1220.200 | 1245.231 | 1220.378 |
| 1 | 2 | 1 | 2 | 1 | 0 | 1323.702 | 1344.561 | 1323.829 |
| 1 | 2 | 1 | 2 | 1 | 1 | 1220.253 | 1245.283 | 1220.431 |
| 1 | 3 | 1 | 0 | 1 | 0 | 1804.955 | 1817.464 | 1805.006 |
| 1 | 3 | 1 | 0 | 1 | 1 | 1521.987 | 1538.665 | 1522.071 |
| 1 | 3 | 1 | 0 | 1 | 2 | 1523.975 | 1544.823 | 1524.102 |
| 1 | 3 | 1 | 1 | 1 | 0 | 1678.454 | 1695.132 | 1678.538 |
| 1 | 3 | 1 | 1 | 1 | 1 | 1523.976 | 1544.824 | 1524.103 |
| 1 | 3 | 1 | 2 | 1 | 0 | 1628.289 | 1649.137 | 1628.416 |
| 1 | 1 | 2 | 0 | 1 | 0 | 1410.802 | 1427.497 | 1410.886 |
| 1 | 1 | 2 | 0 | 1 | 1 | 1118.743 | 1139.612 | 1118.870 |
| 1 | 1 | 2 | 0 | 1 | 2 | 1120.661 | 1145.703 | 1120.838 |
| 1 | 1 | 2 | 1 | 1 | 0 | 1280.183 | 1301.052 | 1280.310 |
| 1 | 1 | 2 | 1 | 1 | 1 | 1120.664 | 1145.706 | 1120.841 |
| 1 | 1 | 2 | 1 | 1 | 2 | 1122.737 | 1151.953 | 1122.974 |
| 1 | 1 | 2 | 2 | 1 | 0 | 1222.756 | 1247.799 | 1222.934 |
| 1 | 1 | 2 | 2 | 1 | 1 | 1122.494 | 1151.710 | 1122.731 |
| 1 | 2 | 2 | 0 | 1 | 0 | 1434.358 | 1451.045 | 1434.442 |
| 1 | 2 | 2 | 0 | 1 | 1 | 1147.844 | 1168.702 | 1147.971 |
| 1 | 2 | 2 | 0 | 1 | 2 | 1150.258 | 1175.288 | 1150.436 |
| 1 | 2 | 2 | 1 | 1 | 0 | 1309.900 | 1330.758 | 1310.027 |
| 1 | 2 | 2 | 1 | 1 | 1 | 1150.888 | 1175.918 | 1151.066 |
| 1 | 2 | 2 | 2 | 1 | 0 | 1249.768 | 1274.798 | 1249.946 |
| 1 | 3 | 2 | 0 | 1 | 0 | 1516.622 | 1533.301 | 1516.707 |
| 1 | 3 | 2 | 0 | 1 | 1 | 1225.311 | 1246.159 | 1225.438 |
| 1 | 3 | 2 | 1 | 1 | 0 | 1382.550 | 1403.398 | 1382.677 |
| 1 | 1 | 3 | 0 | 1 | 0 | 1411.820 | 1432.689 | 1411.946 |
| 1 | 1 | 3 | 0 | 1 | 1 | 1119.269 | 1144.311 | 1119.446 |
| 1 | 1 | 3 | 0 | 1 | 2 | 1121.169 | 1150.386 | 1121.407 |
| 1 | 1 | 3 | 1 | 1 | 0 | 1281.252 | 1306.295 | 1281.429 |
| 1 | 1 | 3 | 1 | 1 | 1 | 1121.173 | 1150.390 | 1121.411 |
| 1 | 1 | 3 | 2 | 1 | 0 | 1224.540 | 1253.757 | 1224.778 |
| 1 | 2 | 3 | 0 | 1 | 0 | 1467.266 | 1488.124 | 1467.392 |
| 1 | 2 | 3 | 0 | 1 | 1 | 1143.565 | 1168.596 | 1143.743 |
| 1 | 2 | 3 | 1 | 1 | 0 | 1337.775 | 1362.805 | 1337.953 |
| 1 | 3 | 3 | 0 | 1 | 0 | 1501.151 | 1521.999 | 1501.278 |
| 1 | 1 | 4 | 0 | 1 | 0 | 1404.794 | 1429.837 | 1404.972 |
| 1 | 1 | 4 | 0 | 1 | 1 | 1121.052 | 1150.269 | 1121.289 |
| 1 | 1 | 4 | 1 | 1 | 0 | 1283.086 | 1312.302 | 1283.323 |
| 1 | 2 | 4 | 0 | 1 | 0 | 1437.959 | 1462.990 | 1438.137 |
| 1 | 1 | 5 | 0 | 1 | 0 | 1415.661 | 1444.877 | 1415.898 |
| 2 | 1 | 0 | 0 | 1 | 0 | 1483.043 | 1495.564 | 1483.094 |
| 2 | 1 | 0 | 0 | 1 | 1 | 1186.255 | 1202.950 | 1186.339 |
| 2 | 1 | 0 | 0 | 1 | 2 | 1188.131 | 1209.000 | 1188.258 |
| 2 | 1 | 0 | 1 | 1 | 0 | 1344.974 | 1361.669 | 1345.058 |
| 2 | 1 | 0 | 1 | 1 | 1 | 1188.131 | 1208.999 | 1188.257 |
| 2 | 1 | 0 | 1 | 1 | 2 | 1190.107 | 1215.149 | 1190.284 |
| 2 | 1 | 0 | 2 | 1 | 0 | 1294.571 | 1315.440 | 1294.698 |
| 2 | 1 | 0 | 2 | 1 | 1 | 1190.128 | 1215.170 | 1190.305 |
| 2 | 1 | 0 | 2 | 1 | 2 | 1192.109 | 1221.325 | 1192.346 |
| 2 | 2 | 0 | 0 | 1 | 0 | 1711.439 | 1723.954 | 1711.489 |
| 2 | 2 | 0 | 0 | 1 | 1 | 1420.433 | 1437.119 | 1420.517 |
| 2 | 2 | 0 | 0 | 1 | 2 | 1422.422 | 1443.281 | 1422.549 |
| 2 | 2 | 0 | 1 | 1 | 0 | 1579.997 | 1596.684 | 1580.081 |
| 2 | 2 | 0 | 1 | 1 | 1 | 1422.422 | 1443.281 | 1422.549 |
| 2 | 2 | 0 | 1 | 1 | 2 | 1424.399 | 1449.430 | 1424.577 |
| 2 | 2 | 0 | 2 | 1 | 0 | 1532.658 | 1553.517 | 1532.785 |
| 2 | 2 | 0 | 2 | 1 | 1 | 1424.409 | 1449.439 | 1424.586 |
| 2 | 3 | 0 | 0 | 1 | 0 | 2042.722 | 2055.230 | 2042.772 |
| 2 | 3 | 0 | 0 | 1 | 1 | 1764.449 | 1781.127 | 1764.533 |
| 2 | 3 | 0 | 0 | 1 | 2 | 1766.211 | 1787.059 | 1766.338 |
| 2 | 3 | 0 | 1 | 1 | 0 | 1925.007 | 1941.686 | 1925.092 |
| 2 | 3 | 0 | 1 | 1 | 1 | 1766.222 | 1787.070 | 1766.349 |
| 2 | 3 | 0 | 2 | 1 | 0 | 1874.001 | 1894.849 | 1874.128 |
| 2 | 1 | 1 | 0 | 1 | 0 | 1410.127 | 1426.822 | 1410.211 |
| 2 | 1 | 1 | 0 | 1 | 1 | 1117.561 | 1138.430 | 1117.687 |
| 2 | 1 | 1 | 0 | 1 | 2 | 1119.461 | 1144.504 | 1119.639 |
| 2 | 1 | 1 | 1 | 1 | 0 | 1279.405 | 1300.274 | 1279.532 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1119.465 | 1144.508 | 1119.643 |
| 2 | 1 | 1 | 1 | 1 | 2 | 1120.457 | 1149.673 | 1120.694 |
| 2 | 1 | 1 | 2 | 1 | 0 | 1222.582 | 1247.625 | 1222.760 |
| 2 | 1 | 1 | 2 | 1 | 1 | 1121.283 | 1150.500 | 1121.521 |
| 2 | 2 | 1 | 0 | 1 | 0 | 1495.823 | 1512.510 | 1495.908 |
| 2 | 2 | 1 | 0 | 1 | 1 | 1204.650 | 1225.509 | 1204.777 |
| 2 | 2 | 1 | 0 | 1 | 2 | 1206.558 | 1231.588 | 1206.736 |
| 2 | 2 | 1 | 1 | 1 | 0 | 1358.898 | 1379.757 | 1359.025 |
| 2 | 2 | 1 | 1 | 1 | 1 | 1206.557 | 1231.587 | 1206.735 |
| 2 | 2 | 1 | 2 | 1 | 0 | 1309.131 | 1334.162 | 1309.309 |
| 2 | 3 | 1 | 0 | 1 | 0 | 1714.820 | 1731.499 | 1714.905 |
| 2 | 3 | 1 | 0 | 1 | 1 | 1429.491 | 1450.339 | 1429.618 |
| 2 | 3 | 1 | 1 | 1 | 0 | 1584.485 | 1605.334 | 1584.613 |
| 2 | 1 | 2 | 0 | 1 | 0 | 1412.086 | 1432.955 | 1412.213 |
| 2 | 1 | 2 | 0 | 1 | 1 | 1119.529 | 1144.571 | 1119.706 |
| 2 | 1 | 2 | 0 | 1 | 2 | 1121.430 | 1150.647 | 1121.667 |
| 2 | 1 | 2 | 1 | 1 | 0 | 1281.374 | 1306.416 | 1281.551 |
| 2 | 1 | 2 | 1 | 1 | 1 | 1121.434 | 1150.651 | 1121.671 |
| 2 | 1 | 2 | 2 | 1 | 0 | 1227.730 | 1256.946 | 1227.967 |
| 2 | 2 | 2 | 0 | 1 | 0 | 1430.807 | 1451.666 | 1430.934 |
| 2 | 2 | 2 | 0 | 1 | 1 | 1141.894 | 1166.924 | 1142.072 |
| 2 | 2 | 2 | 1 | 1 | 0 | 1365.832 | 1390.863 | 1366.010 |
| 2 | 3 | 2 | 0 | 1 | 0 | 1506.043 | 1526.891 | 1506.170 |
| 2 | 1 | 3 | 0 | 1 | 0 | 1401.231 | 1426.274 | 1401.409 |
| 2 | 1 | 3 | 0 | 1 | 1 | 1120.594 | 1149.810 | 1120.831 |
| 2 | 1 | 3 | 1 | 1 | 0 | 1283.342 | 1312.558 | 1283.579 |
| 2 | 2 | 3 | 0 | 1 | 0 | 1433.213 | 1458.244 | 1433.391 |
| 2 | 1 | 4 | 0 | 1 | 0 | 1400.181 | 1429.397 | 1400.418 |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
| NA | NA | NA | NA | NA | NA | NA | NA | NA |
p d q P D Q AIC BIC AICc
214 2 1 1 0 1 1 1117.561 1138.43 1117.687
p d q P D Q AIC BIC AICc
214 2 1 1 0 1 1 1117.561 1138.43 1117.687
p d q P D Q AIC BIC AICc
214 2 1 1 0 1 1 1117.561 1138.43 1117.687
The AIC, AICc, and BIC give the same model SARIMA(2,1,1,0,1,1). SARIMA(1,1, 2,0, 1, 1) also has good performance compared to others.
Cross validation with 1 step ahead forecosts for SARIMA
In this session, a seasonal cross validation using 1 step ahead forecasts and compare the models is performed.
Code
k <- 200 # minimum data length for fitting a model, less than this number will lead to non-stationary fit
n <- length(df.ts.discharge1980)
i=1
err1 = c()
err2 = c()
for(i in 1:(n-k))
{
xtrain <- df.ts.discharge1980[1:(k-1)+i] #observations from 1 to 350
xtest <- df.ts.discharge1980[k+i] #351th observation as the test set
fit <- Arima(xtrain, order=c(2,1,1), seasonal=c(0,1,1))
fcast1 <- forecast(fit, h=1)
fit2 <- Arima(xtrain, order=c(1,1,2), seasonal=c(0,1,1))
fcast2 <- forecast(fit2, h=1)
#capture error for each iteration
# This is mean absolute error
err1 = c(err1, abs(fcast1$mean-xtest))
err2 = c(err2, abs(fcast2$mean-xtest))
# This is mean squared error
err3 = c(err1, (fcast1$mean-xtest)^2)
err4 = c(err2, (fcast2$mean-xtest)^2)
}
(MAE1=mean(err1)) # This is mean absolute error[1] 0.7558021
[1] 0.7619506
[1] 0.9324466
[1] 0.9406275
SARIMA(2,1,1,0,1,1) is the best model based on cross validation, which has lower RMSE of 0.93.
Model diagnosis
Code
Coefficients:
ar1 ar2 ma1 sma1
0.3751 0.1335 -1.0000 -1.0000
s.e. 0.0452 0.0454 0.0111 0.0524
sigma^2 estimated as 0.5251: log likelihood = -553.78, aic = 1117.56
$degrees_of_freedom
[1] 476
$ttable
Estimate SE t.value p.value
ar1 0.3751 0.0452 8.2902 0.0000
ar2 0.1335 0.0454 2.9416 0.0034
ma1 -1.0000 0.0111 -90.2029 0.0000
sma1 -1.0000 0.0524 -19.0976 0.0000
$AIC
[1] 2.328251
$AICc
[1] 2.328427
$BIC
[1] 2.371728
The final model diagnosis (Figure 13) show that p-values of Ljung-box statistic have some pattern left. The ACF of residuals and QQ plot show the residuals are stationary.
Compare with Benchmark methods
Code
fit <- Arima(df.ts.discharge1980, order=c(2,1,1), seasonal=c(0,1,1))
autoplot(df.ts.discharge1980) +
autolayer(meanf(df.ts.discharge1980, h=24),
series="Mean", PI=FALSE) +
autolayer(naive(df.ts.discharge1980, h=24),
series="Naïve", PI=FALSE) +
autolayer(snaive(df.ts.discharge1980, h=24),
series="SNaïve", PI=FALSE)+
autolayer(rwf(df.ts.discharge1980, h=24, drift=TRUE),
series="Drift", PI=FALSE)+
autolayer(forecast(fit,24),
series="fit",PI=FALSE) +
guides(colour=guide_legend(title="Forecast"))From Figure 14, seasonal naive seems have better prediction compared to others. And the fitted SARIMA model has less magnitude of the prediction compared to seasonal naive. In the following, RMSE and MAE are used to compare the performance of the two model.
Summary of Seasonal naive model
ME RMSE MAE MPE MAPE MASE ACF1
Training set -0.006577768 1.172446 0.9249566 -1.322599 12.26289 1 0.4562968
Summary of SARIMA(2,1,1,0,1,1).
Series: df.ts.discharge1980
ARIMA(2,1,1)(0,1,1)[12]
Coefficients:
ar1 ar2 ma1 sma1
0.3751 0.1335 -1.0000 -1.0000
s.e. 0.0452 0.0454 0.0111 0.0524
sigma^2 = 0.5295: log likelihood = -553.78
AIC=1117.56 AICc=1117.69 BIC=1138.43
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 0.02658673 0.7150316 0.5639316 -0.5131699 7.377803 0.6096843
ACF1
Training set 0.0005123345
The statistic shows that the final model SARIMA(2,1,1,0,1,1) has lower RMSE and MAE, which is a better model compared to seasonal naive benchmark model.